Problem of Ages (Problema das Idades)

Question

English: Somebody help me with this challenge? It's very confusing:

Today, both me and my younger brother are between $10$ and $20$ years old. Also, our ages are expressed by prime numbers and the next time this occurs will be in $18$ years. Determine my age knowing that the age of our eldest brother, whose age today is also a prime number, which is one greater than the sum of the ages of me and my younger brother.

Português: Alguém me ajuda com este desafio? É muito confuso:

Eu e meu irmão caçula temos idades entre $10$ e $20$ anos e hoje nossas idades são expressas ambas por números primos, fato que se repetirá pela próxima vez daqui há $18$ anos. Determine minha idade sabendo que a idade de nosso irmão mais velho, que, hoje, também é um número primo, é uma unidade maior do que a soma das nossas idades.

My thoughts: Primes between $10$ and $20$ are $\{ 11, 13,17,19\}$. The primes between $29$ and $37$ ($18$ years later) are $\{ 29,31,37 \}$.

Answer 1

11 and 19: 11+18=29, 19+18=37, 11+19+1=31. No other combination satisfies the condition: 17 is out (17+18=35), 13 is out (11+13+1=25, 13+19+1=33).

Answer 2

Information:

The prime numbers are between 10 and 20: $\{ 11,13,17,19 \}$.

The ages of the two boys are one of those numbers.

Eighteen years after the minimum age of two one turn will $11+18=29$ and the maximum age is $19+18=37$.

The primes between 29 and 37 are: $\{ 29,31,37 \}$.

Remarks:

$$11+18=29$$

$$13+18=31$$

$$17+18=35$$ Is not prime, so the age of any of the boys can be 17.

$$19+18=29$$

Conclusions:

The ages of the boys can be

11 e 13;

11 e 19; or

13 e 19;

Because one is older than the other.

$$11+13+1=25$$ As 25 is not prime, we discard this first possibility;

$$11+19+1=31$$ As 31 is prime, this is a possibility;

$$13+19+1=33$$ As 33 is not prime, we discard this third possibility;

So we have only one possibility which is 11 and 19

Since the problem asks "Determine my age" (age of the oldest, between the two). The final answer is 19 years.